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Hi. I'm online and happy to answer your question today.
You have already done the most important thing: Flushing your eye out with a large amount of clean water.
Just to make sure I understand: It was hair color, not hair bleach, you got in your eye, correct?
Good. OK. You should be OK. This will cause a chemical conjunctivitis but, since your are not in extreme pain and your vision seems OK, I don't think that it has caused a corneal abrasion (chemically removing the skin from the front part of your eye)
I would suggest you use preservative free artificial tears every 2 or 3 hours, or more if you would like, to soothe your eye. There are many brand names of these drops, but they are all about the same so use the least expensive ones. Also use cold compress to help with any swellling.
By tomorrow you should be back to normal.
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